If x^2= 2^x, What is the Value of x ? GMAT Data Sufficiency

Question: If \(x^2=2^x\), what is the value of x ?

\(2x=(\frac{x}{2})^3 \)
\(x=2^{x-2}\)

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are not sufficient.

Correct Answer: D

Solution and Explanation:
Approach Solution 1:
Let us solve this question by evaluating one equation at a time-

Let us solve equation 1 first

\(2x=(\frac{x}{2})^3 \)

\(2x=\frac{x}{8}^3 \)

\(16x=x^3\)

\(x^3-16x=0\)

\(x[x^2-16]=0\)

x [x+4] [x-4] = 0

As you can see,
x can be 0
x can be +4
x can be -4

Since only +4 justifies our question: \(x^2=2^x\), then we can say x = 4

Now, moving on to equation number 2
Let’s substitute \(x=2^{[x-2]}\)in the equation \(x^2=2^x\),

\(2^{[x-2]^2}=2^x\)

\(2^{[2x-4]}=2^x\)

2x - 4 = x

x = 4

The value of x in equation 2 is 4
Since in both equation 1 and equation 2, the value of x is common, which is 4, we can infer that the answer is 4.
Therefore, the value of x is 4 if \(x^2=2^x\).

Hence, each statement alone is sufficient.

Approach Solution 2:
The problem statement states that
Given:

\(x^2=2^x\)

Find out:

The value of x.

Statement 1: \(2x=(\frac{x}{2})^3 \)
\(x*(\frac{x^2}8 - 2) = 0 \)
we get x as 0, 4 or -4.
When we substitute value in \(x^2=2^x\), we get the ONLY possible value of x = 4.
Hence, statement 1 alone is sufficient.

Statement 2: \(x=2^{[x-2]}\)
or, \(x=2^{x-2}\)
Therefore, 2^x has to be a multiple of 2 and hence also x should be in the power of 2. Therefore, it is 2,4,8,16… and x will fit in only at x = 4 as \(2^{[4-2]}\)
ONLY when x is 4 both sides are 4.
Hence, statement 2 alone is sufficient

Thus, each statement alone is sufficient.

Approach Solution 3:
The problem statement informs that
Given:

\(x^2=2^x\)

Find out:

The value of x.

This question turns out to state there are only two possible cases:
Case i: x=2
Case ii: x=4
In other words, the given information is telling us that x is either 2 or 4.
This approach is super easy to solve the problem. Let’s check each statement.

Statement 1: \(2x=(\frac{x}{2})^3 \)
Let's just test the two possible x-values.
Test x=2, to get: \(2(2)=(\frac{2}{2})^3 \), which can be simplified to get: 4=1. Hence it is FALSE.
Therefore, x≠2, which means it must be the case that x=4
Since we can answer the question with certainty.

Hence, statement 1 is SUFFICIENT

Statement 2: \(x=2^{[x-2]}\)
Once again, let’s test the two possible values of x.
Test x=2, to get: \(2=2^{[2-2]}\), which can be simplified to get: 2=1. Hence it is FALSE.
Therefore, x≠2, which means it must be the case that x=4
Since we can answer the question with certainty.
Hence, statement 2 is SUFFICIENT

Thus, each statement alone is sufficient.

“If \(x^2=2^x\), what is the value of x”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This question has been taken from the book “GMAT Prep Plus”. The GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions include a problem statement that is followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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If x^2= 2^x, What is the Value of x ? GMAT Data Sufficiency

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